plot::Polygon2d – 2D polygons

plot::Polygon3d – 3D polygons

plot::Polygon2d, plot::Polygon3d define polygons in 2D and 3D, respectively, by a given list of vertex points.

→ Examples

Calls:

plot::Polygon2d([pt2d1, pt2d2, ...], <a = amin .. amax>, Options)

plot::Polygon3d([pt3d1, pt3d2, ...], <a = amin .. amax>, Options)

plot::Polygon2d( $`M_{2d}`$ , <a = amin .. amax>, Options)

plot::Polygon3d( $`M_{3d}`$ , <a = amin .. amax>, Options)

Parameters:

pt2d1, pt2d2, ...:	the 2D vertices. These must not be of type plot::Point2d, but lists of two numerical real values or arithmetical expressions of the animation parameter a (the coordinates). pt2d1, pt2d2, ... is equivalent to the attribute Points2d.
pt3d1, pt3d2, ...:	the 3D vertices. These must not be of type plot::Point3d, but lists of three numerical real values or arithmetical expressions of the animation parameter a (the coordinates). pt3d1, pt3d2, ... is equivalent to the attribute Points3d.
$`M_{2d}`$ :	an array or a matrix with 2 columns. Each row provides the coordinates of one point. $`M_{2d}`$ is equivalent to the attribute Points2d.
$`M_{3d}`$ :	an array or a matrix with 3 columns. Each row provides the coordinates of one point. $`M_{3d}`$ is equivalent to the attribute Points3d.

Details:

A polygon consists of points and egdes. The edges are made up of the lines traversing from the first to the second point, the second to the third point, and so on. The last point is automatically connected with the first point if the attribute Closed = TRUE is specified.
Points and lines can be hidden via PointsVisible = FALSE and LinesVisible = FALSE. Per default the vertex points are hidden while the edges are visible.
All points as a whole can be manipulated via PointStyle and PointSize. The attribute LineColor sets the color for all points and all lines. Likewise all lines can be manipulated via LineStyle and LineWidth.
It is possible to vary the color of all lines and points via LineColorType. The default value is Flat. Specifying the values Dichromatic or Rainbow, a second color LineColor2 can be set. With Functional, the colors are taken from a user defined LineColorFunction. Cf. example 2.
The area of any closed 2D polygon can be filled by specifying Filled = TRUE. The filled area is defined by connecting the last and the first vertex. This additional edge itself, however, is only displayed if Closed = TRUE is set.

A fill color and a fill pattern can be chosen by FillColor and FillPattern.

In case of a self-intersecting polygon, a FillStyle can be selected. Cf. example 3.
A 3D polygon can only be filled if it is defined by 3 vertices (a triangle). The attribute Filled = TRUE is ignored for other 3D polygons.

Example 1

We create a polygon with vertices located at the complex roots of . The polygon consists of the lines joining the points in the order given:

p := plot::Polygon2d(
[[cos(2*PI*k/5), sin(2*PI*k/5)] $ k = 0..4]):
plot(p, plot::Circle2d(1, [0, 0])):

MuPAD graphics

In order to include the line connecting the last with the first point, pass the attribute Closed to the polygon:

p::Closed := TRUE:
plot(p, plot::Circle2d(1, [0, 0])):

MuPAD graphics

delete p

Example 2

We plot a closed star-shaped 3D polygon with various color attributes:

p := plot::Polygon3d(
       [([cos(PI*k/3), sin(PI*k/3), sin(PI*k/3)],
         [cos(PI*k/3 + PI/6)/2,
          sin(PI*k/3 + PI/6)/2,
          sin(PI*k/3 + PI/6)/2]) $k = 1..6
       ], Closed = TRUE):
S1 := plot::Scene3d(p, LineColorType = Flat):
S2 := plot::Scene3d(p, LineColorType = Dichromatic):
S3 := plot::Scene3d(p, LineColorType = Dichromatic,
                    LineColor = RGB::Blue,
                    LineColor2 = RGB::Green):
S4 := plot::Scene3d(p, LineColorType = Rainbow,
                    LineColor = RGB::Blue,
                    LineColor2 = RGB::Green):
plot(S1, S2, S3, S4)

MuPAD graphics

We plot the same polygon while animating its line color using a color function. The result is a dazzling star:

p := plot::Polygon3d(
       [([cos(PI*k/3), sin(PI*k/3), sin(PI*k/3)],
         [cos(PI*k/3 + PI/6)/2,
          sin(PI*k/3 + PI/6)/2,
          sin(PI*k/3 + PI/6)/2]) $k = 1..6
       ], Closed = TRUE,
       LineColorFunction =
           proc(x, y, z, i, a) begin
             [sin(x + a*i)^2, sin(y + a*i)^2, sin(z + a*i)^2]:
           end_proc,
       a = 0..10):
plot(p)

MuPAD graphics

delete p, S1, S2, S3, S4

Example 3

We plot a closed 2D polygon and fill the area inside. In fact, there are two possible interpretations of what “inside” really means. In the first plot, the complement of the unbound component of the complement of the polygon is filled. In the second plot only that area is filled that contains points with non-zero winding number with respect to the polygon. Cf. FillStyle for a detailed discussion.

p := plot::Polygon2d(
      [[cos(PI*k/3), sin(PI*k/3)] $k = 1..6,
       [cos(PI*k/3 + PI/6)/2, sin(PI*k/3 + PI/6)/2] $k = 1..6
      ], Closed = TRUE):
S1 := plot::Scene2d(p, Filled = TRUE):
S2 := plot::Scene2d(p, Filled = TRUE, FillStyle = Winding):
plot(S1, S2, Layout = Horizontal, Axes = Frame,
     Scaling = Constrained)

MuPAD graphics

delete p, S1, S2:

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